Average Error: 16.0 → 15.3
Time: 48.5s
Precision: 64
Internal precision: 128
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
⬇
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}} \le -1.6242936666275148 \cdot 10^{+39}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}} \le -1.4475800306973208 \cdot 10^{-69}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}} \le 0.0:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original16.0
Comparison13.3
Herbie15.3
\[ \begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t \lt 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes.

  2. if (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -1.6242936666275148e+39 or 0.0 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))

    1. Initial program 8.8

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied div-inv 8.9

      \[\leadsto \color{blue}{\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}}\]

    if -1.6242936666275148e+39 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < -1.4475800306973208e-69

    1. Initial program 0.3

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied associate-/l* 3.0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \color{blue}{\frac{y}{\frac{t}{b}}}}\]

    if -1.4475800306973208e-69 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 0.0

    1. Initial program 29.3

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied associate-/l* 26.6

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \color{blue}{\frac{y}{\frac{t}{b}}}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 48.5s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3315119702 2796268412 2600842881 3543422056 3318086707 631567159)'
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))